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Chicago Public Schools, Advanced Algebra Lesson plans, Days 37-53

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<ul><li><p>134</p><p>STRUCTURED CURRICULUM LESSON PLAN</p><p>Day: 037 Subject: Advanced Algebra Grade Level: 11</p><p>Correlations (SG,CAS,CFS): 6A1</p><p>TAP:Analyze and interpret data presented in charts,</p><p>graphs, tables and other displays</p><p>ISAT:Identify, analyze, and solve problems using</p><p>equations, inequalities, functions, and theirgraphs</p><p>Unit Focus/Foci</p><p>Matrices and Systems of Equations</p><p>Instructional Focus/Foci</p><p>Using Matrices to Store and Represent Data and Adding or Subtracting Matrices</p><p>Materials</p><p>Classroom set of graphing calculatorsOverhead graphing calculatorOverhead projector</p><p>Educational Strategies/Instructional Procedures</p><p>Give the following definitions:</p><p>Matrix: A rectangular arrangement of elements in rows and columns enclosed by brackets.</p><p>Example: M = 1 4 3 2</p><p>3 2 0 5</p><p>LNM</p><p>OQP</p><p>Matrix notation: An entry is named as mab, where a = row number and b = column number</p><p>In the example above, m13= 3; m24=5Dimension: the number of rows x the number of columns gives the dimension of amatrix. The dimension of the matrix M above is 2 x 4.</p><p>Have students place elements in their proper position, then have them enter the followingmatrices into their calculators:</p></li><li><p>135</p><p>To enter a matrix into the graphing calculator:1. press MATRX2. use the right arrow to EDIT, then ENTER3. enter number of rows, ENTER4. enter number of columns, ENTER5. enter each element, going across. After each element, press ENTER. If you</p><p>make a mistake, scroll to the location you want to change.</p><p>To perform operations on matrices, press MATRX, leave NAMES highlighted and scroll downto the matrix you want to use; press ENTER.</p><p>Students will find: A + B; B A; A A (call this C); C + B; 1 x B</p><p>Students will report findings to the whole class.</p><p>Integration with Core Subject(s)</p><p>LA: Understand explicit, factual informationUnderstand the meaning of words in context</p><p>Connection(s)</p><p>Enrichment: In a recent survey, 90% of high school students were opposed to a systemwideSlice Girls concert suggested by elementary school students.1. Work with a partner to conduct a survey in your school regarding the questions below.</p><p>A. Should the Slice Girls concert occur?B. Should high school students be mandated to attend?C. Should elementary school students be mandated to attend?D. What percent have no opinion?</p><p>Use the following matrix to report your data in percentage form.</p><p>Question Yes No Indifferent</p><p>A B=</p><p>L</p><p>NMMM</p><p>O</p><p>QPPP</p><p>=</p><p>L</p><p>NMMM</p><p>O</p><p>QPPP</p><p>3 2 1 0</p><p>6 2 4 1</p><p>0 3 2 1</p><p>7 3 2 1</p><p>4 0 3 2</p><p>1 7 3 0</p><p>ABCD</p></li><li><p>136</p><p>Fine Arts:</p><p>Home: Parents will sign homework record weekly.</p><p>Remediation:</p><p>Technology:</p><p>Assessment</p><p>Have students answer the following questions:</p><p> a. What must be true about the dimensions of two matrices that are equal? b. Given matrices A, B, and A + B, why must they have the same dimension? c. What is the relationship between matrices A and 1A? d. What is a good name for matrix C?</p><p>Homework</p><p>Assign appropriate problems from your text.</p><p>Teacher Notes</p><p>Solutions to Assessment:</p><p>a. They must have the same number of rows and columns.b. The entries from the first matrix should correspond to the entries from the second matrix.c. Matrix -1 x A is the opposite of matrix A.d. Zero matrix</p><p>Solutions to in-class assignment:</p><p>A + B = </p><p>11101</p><p>17210</p><p>11110</p><p>B A = </p><p>1541</p><p>3122</p><p>1354</p><p>A A = </p><p>0000</p><p>0000</p><p>0000</p><p>C + B = </p><p>0371</p><p>2304</p><p>1237</p><p>-1 x B</p><p>0371</p><p>2304</p><p>1237</p></li><li><p>137</p><p>STRUCTURED CURRICULUM LESSON PLAN</p><p>Day: 038 - 039 Subject: Advanced Algebra Grade Level: High School</p><p>Correlations (SG,CAS,CFS): 6A1</p><p>TAP:Perform arithmetic operations involving</p><p>integers, fractions, decimals and percentsexplicitly stated or within context</p><p>Choose and apply appropriate operationalprocedures and problem-solving strategies toreal world situations</p><p>Understand number systemsUse variables, number sentences, and equations</p><p>to represent solutions and solve problems</p><p>ISAT:Solve problems requiring computations with</p><p>whole numbers, fractions, decimals, ratios,percents, and proportions</p><p>Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results</p><p>Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs</p><p>Unit Focus/Foci</p><p>Exploring Matrices and Systems of Equations</p><p>Instructional Focus/Foci</p><p>Performing Matrix Multiplication</p><p>Materials</p><p>Classroom set of graphing calculatorsOverhead graphing calculatorsClassroom set of graphing calculatorsOverhead projector</p><p>Educational Strategies/Instructional Procedures</p><p>Explain to students: To multiply two matrices Amxn and Bpxq, n must equal p (the inner</p><p>dimensions). The resulting matrix AB will have dimension m x q (the outer dimensions). If np, the two matrices cannot be multiplied.</p><p>Example 1: A B=</p><p>LNM</p><p>OQP = </p><p>L</p><p>N</p><p>MMMM</p><p>O</p><p>Q</p><p>PPPP3 2 0 1</p><p>4 3 3 2</p><p>0</p><p>3</p><p>2</p><p>1</p></li><li><p>138</p><p>dim[A] = 2x4, dim[B] = 4x1, so these two matrices can be multiplied (since 4=4), and theresulting matrix will have dimension 2x1.</p><p>Take row 1 of A and column 1 of B, multiply successive elements, add these products, and theresult is the corresponding element in the product matrix. Using the example above:</p><p>30 + 23 + 02 + 11 = 5; 40 +03 + 32 + -21 = 4. So the product matrix is 5</p><p>4</p><p>LNMOQP</p><p>Give students several matrices of different dimensions and ask if they can be multiplied.</p><p>Example 2:</p><p>a. </p><p>40</p><p>36</p><p> 25</p><p>68b. </p><p>82</p><p>43</p><p>96</p><p>41</p><p>72c. </p><p>036</p><p>894</p><p>282</p><p>413</p><p>Solutions: a. yes b. yes c. no</p><p>Have students form small groups; use the graphing calculator to determine whether theassociative, commutative, or distributive properties hold for matrix multiplication.</p><p>Report findings to class. Use the matrices: A = </p><p> 06</p><p>14 B = </p><p>63</p><p>12 C = </p><p>20</p><p>31</p><p>a. A*B = B * Ab . A(B*C) = (A * B ) * Cc . A(B+C) = AB + AC</p><p>Example 3:</p><p>Jane and Andrew enjoy shopping for music on a monthly basis. They each purchase thefollowing items.</p><p>CDs Tapes LPsJane 10 6 1</p><p>Andrew 8 2 3</p><p>If CDs cost $15, tapes cost $8, and LPs cost $10, use matrices to find the total cost that Jane andAndrew each spent on music.</p></li><li><p>139</p><p>Solution: </p><p>328</p><p>1610</p><p>10</p><p>8</p><p>15</p><p>= </p><p>166.</p><p>208</p><p>Integration with Core Subject(s)</p><p>LA: Understand explicit, factual informationUnderstand the meaning of words in context</p><p>Connection(s)</p><p>Enrichment: Use graph paper to draw squares with areas of 2, 5, and 10 units.</p><p>Fine Arts:</p><p>Home: Parents will sign homework record weekly.</p><p>Remediation:</p><p>Technology: Have students review how their calculator can perform various operations withmatrices.</p><p>Assessment</p><p>Teacher observation</p><p>Homework</p><p>Assign appropriate problems from your text.</p><p>Teacher Notes</p><p>This is a two-day lesson. The second day (Day 39) should be used to review homework, forremediation exercises, or for extra practice, as necessary.</p><p>Solutions to in-class assignment:</p><p>a. A*B = B *A (not true) b. A(B*C) = (A *B) * C (true)</p><p> 612</p><p>25</p><p>=</p><p>348</p><p>214</p><p>2412</p><p>195= </p><p>2412</p><p>195</p><p>c. A(B+C) = AB + AC (not true)</p><p> invalid dimensions = </p><p>246</p><p>121</p></li><li><p>140</p><p>STRUCTURED CURRICULUM LESSON PLAN</p><p>Day: 040 Subject: Advanced Algebra Grade Level: High School</p><p>Correlations (SG,CAS,CFS): 8D2</p><p>TAP:Perform arithmetic operations involving</p><p>integers, fractions, decimals and percentsexplicitly stated or within context</p><p>Choose and apply appropriate operationalprocedures and problem-solving strategies toreal world situations</p><p>Understand number systemsUse variables, number sentences, and equations</p><p>to represent solutions and solve problems</p><p>ISAT:Solve problems requiring computations with</p><p>whole numbers, fractions, decimals, ratios,percents, and proportions</p><p>Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results</p><p>Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs</p><p>Unit Focus/Foci</p><p>Exploring Matrices and Systems of Equations</p><p>Instructional Focus/Foci</p><p>Identifying Systems of Two Linear Equations</p><p>Materials</p><p>Handout 4.3Classroom set of graphing calculatorsOverhead graphing calculatorOverhead projector</p><p>Educational Strategies/Instructional Procedures</p><p>Students will: Form groups of 2. Complete Handout 4.3 (exploration on intersections of lines). Report findings to the class.</p><p>Present the following definitions:</p><p>Consistent system: lines intersect in one or more pointsIndependent system: lines intersect in one unique point (one solution)Dependent system: lines intersect in an infinite number of points (the lines coincide; infinitelymany solutions)Inconsistent system: lines do not intersect (parallel lines; no solutions)</p></li><li><p>141</p><p>Give examples of systems of equations and graphs of solutions of equations.</p><p>Have students match each system with its graph.</p><p>a. y = x + 1 b. y = -2x + 3 c. y = .5x - 2 y = x y = 3x + 1 y = -x + 4</p><p> 1. 2. 3..</p><p>Solutions: a 3 b 1 c 2</p><p>Integration with Core Subject(s)</p><p>LA: Understand explicit, factual informationUnderstand the meaning of words in context</p><p>Connection(s)</p><p>Enrichment: Have students discuss how to solve a system of three linear equations in threevariables.</p><p>Fine Arts:</p><p>Home: Parents will sign homework record weekly.</p><p>Remediation:</p><p>Technology: Students will review the various ways their calculator can graph and find theintersections of linear functions.</p><p>Assessment</p><p>Describe the three types of systems of linear equations in terms of intersecting lines.</p><p>Homework</p><p>Assign appropriate problems from your text.</p></li><li><p>142</p><p>Teacher Notes</p><p>Solutions to Handout 4.3:</p><p>Type 1: a. Intersects at (1, 2) b. Intersects at c. Intersects at (-7, 1)</p><p>Type 2: a. Always intersect b. Always intersect c. Always intersect</p><p>Type 3: a. Parallel b. Parallel c. Parallel</p><p> FHIK</p><p>5</p><p>8</p><p>1</p><p>2,</p></li><li><p>143</p><p>Handout 4.3</p><p>This exploration involves different types of systems of equations and their graphs. For each type,graph each pair of equations and the same coordinate plane. Then answer the questions.</p><p>Type 1:</p><p>1.x 2y 5</p><p>3x 2y 72.</p><p>4x 3y 1</p><p>4x y 33.</p><p>x 2y 5</p><p>x 3y 4</p><p>+ =</p><p>+ =</p><p>+ =</p><p> =</p><p> =</p><p> + =</p><p>RSTUVW</p><p>RSTUVW</p><p>RSTUVW</p><p>Which pairs of lines intersect?How is each pair of lines alike or different?</p><p>Type 2:</p><p>1.x y 4</p><p>3x 3y 122.</p><p>x 2y 6</p><p>4x 8y 243.</p><p>4x y 1</p><p>12x 3y 3</p><p>+ =</p><p>+ =</p><p> =</p><p> + =</p><p>+ = </p><p> =</p><p>RSTUVW</p><p>RSTUVW</p><p>RSTUVW</p><p>Which pairs of lines intersect?How is each pair of lines alike or different?</p><p>Type 3:</p><p>1.x 3y 15</p><p>x 3y 62.</p><p>2x y 13</p><p>4x 2y 143.</p><p>x 4y 1</p><p>x 4y 2</p><p> = </p><p> = </p><p> + = </p><p> + =</p><p>+ =</p><p> = </p><p>RSTUVW</p><p>RSTUVW</p><p>RSTUVW</p><p>Which pairs of lines intersect?How is each pair of lines alike or different?</p><p>Describe the three types of systems of linear equations in terms of intersecting lines.</p></li><li><p>144</p><p>STRUCTURED CURRICULUM LESSON PLAN</p><p>Day: 041 Subject: Advanced Algebra Grade Level: High School</p><p>Correlations (SG,CAS,CFS): 8D2</p><p>TAP:Perform arithmetic operations involving</p><p>integers, fractions, decimals and percentsexplicitly stated or within context</p><p>Choose and apply appropriate operationalprocedures and problem-solving strategies toreal world situations</p><p>Understand number systemsUse variables, number sentences, and equations</p><p>to represent solutions and solve problems</p><p>ISAT:Solve problems requiring computations with</p><p>whole numbers, fractions, decimals, ratios,percents, and proportions</p><p>Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results</p><p>Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs</p><p>Unit Focus/Foci</p><p>Exploring Matrices and Systems of Equations</p><p>Instructional Focus/Foci</p><p>Solving a System of Two Linear Equations Algebraically</p><p>Materials</p><p>Classroom set of graphing calculators</p><p>Educational Strategies/Instructional Procedures</p><p>Present the following situational problem: Teacher will tell students: I am thinking of twonumbers whose sum is 30 and whose difference is 10. What are the two numbers? (20, 10) Havestudent volunteers make up additional problems like this one for class to solve.</p><p>Ask students how to write a pair of equations to represent the problem.</p><p>x + y = 30x y = 10</p><p>Ask: What happens if we add the two equations? [The ys eliminate and we get2x = 40 x = 20.] How can we then find y? [Replace into one of the equations: y = 10.]</p><p>This is an example of solving a system of equations using elimination by addition.</p></li><li><p>145</p><p>Example 1:3x + 2y = 8 _ 3x + 2y = 8 2(2) + y = 52x + y = 5 4x 2y = 10 4 + y = 5</p><p> 1x = 2 y = 1x = 2 so, x = 2, y = 1</p><p>Example 2: 5x y = 6 -y = -5x+6 y = 5x-6 y = 5x-6</p><p> 0 =0</p><p>Example 3: x +y = 7 2x + 2y =14 2x + 2y =4 2x + 2y = 4</p><p>0 10no solution</p><p>.1. Different slope Lines intersect One solution Independent system2. Same slope andintercepts</p><p>Lines coincide Infinite number ofsolutions</p><p>Dependent system</p><p>3. Same slope anddifferent intercepts</p><p>Lines parallel No solution Inconsistent system</p><p>Cooperative Learning: Separate the class into pairs of students. Assign a system of equations andhave one student of the pair solve the system using the elimination method while the other solvesthe system using a graphing calculator and the TRACE feature. Have the pairs compare theiranswers and then share them with the class. Have pairs exchange roles and repeat.</p><p>Integration with Core Subject(s)</p><p>LA: Understand explicit, factual informationUnderstand the meaning of words in context</p><p>Connection(s)</p><p>Enrichment:</p><p>Fine Arts:</p><p>Home: Parent will sign homework record weekly.</p><p>Remediation:</p><p>Technology: </p></li><li><p>146</p><p>Assessment</p><p>Assign students a system of equations and have them describe the steps for solving them.</p><p>Homework</p><p>Assign appropriate problems from your text.</p><p>Teacher Notes</p><p>The cooperative learning activities must be monitored very closely.Prepare copies of Solving Systems Algebraically.</p></li><li><p>147</p><p>STRUCTURED CURRICULUM LESSON PLAN</p><p>Day: 042 Subject: Advanced Algebra Grade Level: High School</p><p>Correlations (SG,CAS,CFS): 8D2</p><p>TAP:Perform arithmetic operations involving</p><p>integers, fractions, decimals and percents,explicitly stated or within context</p><p>Understand number systemsUse variables, number sentences, and equations</p><p>to represent solutions and solve problems</p><p>ISAT:Solve problems requiring computations with</p><p>whole numbers, fractions, decimals, ratios,percents, and proportions</p><p>Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results</p><p>Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs</p><p>Unit Focus/Foci</p><p>Matrices and Systems of Equations</p><p>Instructional Focus/Foci</p><p>Solving a System of Two Linear Equations Algebraically</p><p>Materials</p><p>Copies of Solving Systems Algebraically</p><p>Educational Strategies/Instructional Procedures</p><p>Substitution is the second algebraic method used to solve a system of equations. One equation issolved for one variable in terms of the other. This expression for the variable is substituted in theother equation.</p><p>Example 1: 3x + 2y = 9 y = x + 2</p><p>3x +2(x +2) = 9 y= x+2 3x+2x+4 = 9 y = 1+2</p><p>5x+4 = 9 y = 35x= 5x = 1</p></li><li><p>148</p><p>Example 2: x 3y = -122x + 11y = -6</p><p> x = 3y 12 2(3y 12) + 11y = -6 x3 = -12 6y 24 +11y = -6 x = -9</p><p>18y 24 = -618y = 18 y = 1</p><p>Have the students complete Solving Systems Algebraically.</p><p>Integration with Core Subject(s)</p><p>LA: Understand explicit, factual informationUnderstand the meaning of words in context</p><p>SC: Understand uses of scientific units and instrumentsApply scientific method to solve problems</p><p>Conne...</p></li></ul>

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